How to set weights for a mesh to bend smoothly?
From your image, it looks like you don't have any weights painted on your root bone (i.e. the one for the blue part on the left). Adding these weights (and possibly the bone if it isn't there) should allow you to adjust how smooth the blend is.

In order to be able blend between bones, blender has to actually know which bones to blend. To do this it looks at all the bones that have some weight assigned to a vertex and blends the effect of the bones based on the relative strengths of those weights. If a vertex has only one bone weighted to it, regardless of the value of that weight, it will only be influenced by that bone.